Now, the objective is to apply LaSalle's invariance principle in order
to prove the asymptotic stability of the observer error, where the
proof follows the intention of [1]}. However, the
applicability of LaSalle's invariance principle according to [2]}
requires the precompactness of the solution trajectories, which is
not ensured in the infinite-dimensional scenario. Since in the previous
section we have shown that \(\mathcal {A}^{-1}\)  is bounded, by means
of the Sobolev embedding theorem, it follows that \(\mathcal {A}^{-1}\) 
is compact (see proof of Lemma 2.4 in [3]} or [2]}),
which further implies the precompactness of the trajectories, see
[5]}.
